A voter model with concealed and publicly expressed opinions

The voter model is a simple agent-based model to mimic opinion dynamics in social networks: a randomly chosen agent adopts the opinion of a randomly chosen neighbour. This process is repeated until a consensus emerges. Although the basic voter model is theoretically intriguing, it misses an important feature of real opinion dynamics: it does not distinguish between an agent's publicly expressed opinion and her inner conviction. A person may not feel comfortable declaring her conviction if her social circle appears to hold an opposing view [1]. Here we introduce the Concealed Voter Model where we add a second, concealed layer of opinions to the public layer. We study a complete graph of agents who can choose from two opinions. In Figure 1, we represent each agent by two nodes: one in the public and one in the concealed layer. The agent marked by a yellow background has a different opinion in the two layers (externally red, internally blue). The following three elementary events can happen to this agent. (1) She copies the public opinion of a randomly selected neighbour. (2) She discloses her previously secret point of view (“externalization”). Or (3) she accepts her public opinion as inner conviction (“internalization”). The events (1)−(3) happen randomly with rates c, e and i, respectively.
We define a martingale M that determines the probability of all agents eventually agreeing on a particular opinion. By analyzing the evolution of M in the limit of a large number of agents, we derive the leading-order terms for the mean and standard deviation of the consensus time (i.e. the time needed until all opinions are identical). We thereby give a precise prediction by how much concealed opinions slow down a consensus.